Thermo & Stat Mech - Spring 2006 Class 15

1

Thermodynamics and Statistical Mechanics

Transport Processes

Thermo & Stat Mech - Spring 2006 Class 15

2

Mean Free Path

The average distance that a particles travels between collisions is called the mean free path, . In order to have a collision, the centers of two particles of radius R must approach to within a distance of 2R = d. The picture on the next slide illustrates this point.

Thermo & Stat Mech - Spring 2006 Class 15

3

Collision Cross Section

d2

Thermo & Stat Mech - Spring 2006 Class 15

4

Mean Free Path

)/(

1

1 so ,1

vvnv

vnvn

rel

relrel

Thermo & Stat Mech - Spring 2006 Class 15

5

Mean Free Paths(?)

nn

nn

nn

171.0

1

2

1

177.0

1

4

3

175.0

1

4

3

Thermo & Stat Mech - Spring 2006 Class 15

6

Mean Free Path

mc

mmrelm

relrel

vf

nn

vvnvvnv

vnvn

1163.0

8

1

)/2(

1

)/(

1

1 so ,1

Thermo & Stat Mech - Spring 2006 Class 15

7

Some Molecular Speeds

ProbableMost 2

3

3)(

8)(

max

2

0

22

0

m

kTv

m

kTvv

m

kTdvvfvv

m

kTdvvvfv

rms

Thermo & Stat Mech - Spring 2006 Class 15

8

Effusion

If there is a hole in a container of gas, molecules will escape through it. The rate at which they pass through the hole is equal to the flux of particles hitting the hole times the area of the hole, A. Then the rate of change of the number of molecules in the container is:

Adt

dN

Thermo & Stat Mech - Spring 2006 Class 15

9

Effusion Limit

AvnAdt

dNvn

DD

DD

41

41

22

2222

and Then,

4.0or 16.0 i.e.

16or 44

If

If the hole is small equilibrium conditions prevail, and is known.

Thermo & Stat Mech - Spring 2006 Class 15

10

Transport Processes

1. Molecular diffusion. The movement of the particles is of interest. This is particle transport.

2. Thermal conductivity. Particles carry energy from a high temperature region to a lower temperature region. This is energy transport.

3. Viscosity. Drag is created when particles move to a region which is moving at a different speed. This is momentum transport.

Thermo & Stat Mech - Spring 2006 Class 15

11

Transport Processes

dy

dv

A

F

dy

dT

dt

dQ

A

dy

ndD

xx

y

.3

1.2

.1

Thermo & Stat Mech - Spring 2006 Class 15

12

Viscosity

Consider a plate moving parallel to another plate with a layer of gas between them. The gas in contact with either plate is at rest relative to the plate, so there is a velocity gradient in the gas. A force F is applied to each plate to maintain the motion, and the area of the upper plate is A.

Thermo & Stat Mech - Spring 2006 Class 15

13

Viscosity

Thermo & Stat Mech - Spring 2006 Class 15

14

Viscosity

Gas diffuses up and down, transferring x momentum from layer to layer. The gas that moves in the + y direction is going slower than the layer it moves into, and the gas that moves in the – y direction is going faster than the layer it moves into.

Thermo & Stat Mech - Spring 2006 Class 15

15

Viscosity

Thermo & Stat Mech - Spring 2006 Class 15

16

Viscosity

First we need to know how far particles travel between collisions in the y direction. We must find the average value of y = cos , averaged over the flux of molecules.

dddvvvfn

d cossin])([4

Thermo & Stat Mech - Spring 2006 Class 15

17

Viscosity

3

2)2(

3

1)(

41

cossin)(4

cos

2

0

2

0

2

0

vv

y

vn

dddvvvfn

y

dy

Thermo & Stat Mech - Spring 2006 Class 15

18

Viscosity

(poise) 31

31

31

34

41

32

41

32

41

mvndy

dvmvn

A

FG

dy

dvmvn

dy

dvmvnGGG

dy

dvvmvnG

dy

dvvmvnG

xx

xx

xx

xx

Thermo & Stat Mech - Spring 2006 Class 15

19

Viscosity

2

31

31

33

8

18

8

18

d

mkTmkT

n

mm

kTn

nm

kTvmvn

Thermo & Stat Mech - Spring 2006 Class 15

20

Diffusion

vD

dy

ndv

dy

ndvGG

dy

ndynvG

dy

ndynvG

31

31

34

41

32

41

32

41

)(

)(

Thermo & Stat Mech - Spring 2006 Class 15

21

Thermal Conductivity

A

v

N

Tc

dy

dvn

dy

dvnGG

dt

dQ

A

dy

dyvnG

dy

dyvnG

31

34

41

32

41

32

41

1

)(

)(

Thermo & Stat Mech - Spring 2006 Class 15

22

Thermal Conductivity

A

v

A

v

A

v

A

v

N

cvn

dy

dT

N

cvn

dy

dvn

dt

dQ

A

dy

dT

N

c

dy

dT

dT

d

dy

d

N

Tc

31

31

311

Thermo & Stat Mech - Spring 2006 Class 15

23

Transport Processes

mvndy

dv

A

F

N

cvn

dy

dT

dt

dQ

A

vDdy

ndD

xx

A

v

y

31

31

31

.3

1.2

.1